Friday, September 2, 2011

Fraunhofer Diffraction

Last week’s blog entry discussed Fresnel diffraction, which Russ Hobbie and I analyzed in the 4th edition of Intermediate Physics for Medicine and Biology when we examined the ultrasonic pressure distribution produced near a circular piezoelectric transducer. This week, I will analyze diffraction far from the wave source, known as Fraunhofer diffraction, named for the German scientist Joseph von Fraunhofer.

The mathematics of Fraunhofer diffraction is a bit too complicated to derive here, but the gist of it can be found by inspecting the first equation in Section 13.7 (Medical Uses of Ultrasound), found at the bottom of the left column on page 351. The pressure is found by integrating 1/r times a cosine function over the transducer face. When you are far from the transducer, r is approximately a constant and can be taken out of the integral. In that case, you just integrate cosine over the transducer area. This becomes similar to the two-dimensional Fourier transform defined in Chapter 12 (Images). In fact, the far field pressure distribution produced by a circular transducer given in Eq. 13.40 is the same Bessel function result as derived in Problem 10 of Chapter 12.

The intensity distribution in Eq. 13.40 is known as the Airy pattern, after the English scientist and mathematician George Biddell Airy. As shown in Fig. 13.15, the pattern consists of a central peak, surrounded by weaker secondary maxima. The Airy pattern occurs during imaging using a circular aperture, such as when viewing stars through a telescope. Two adjacent stars appear as two Airy patterns. Distinguishing the two stars is difficult unless the separation between the images is greater than the separation between the peak of the Airy pattern and its first zero. This is called the Rayleigh criterion, after Lord Rayleigh. Rayleigh (1842-1919, born John William Strutt)--one of those 19th century English Victorian physicists I like so much--did fundamental work in acoustics, and published the classic textbook Theory of Sound.

1. Is the Airy pattern what we observe when looking at a distant streetlamp through our circular aperature, the pupil?

2. Diffraction does contribute to the resolution of the eye (see page 390 of our book). However, I do not think it has a lot to do with what you perceive when looking at a streetlight. If it was simply a diffraction effect, then viewing the distance stars would have the same effect. My understanding is the the distance from the center to the first minimum of the Airy pattern is similar to the separation between photoreceptors in the eye.

3. I heard recently that there has been a new star discovered in Leo, within the Milky Way. Named SDS J102915+172927, it apparently 'should not be there' because it is does not consist of the elements that astronomers expected necessary for such low mass stars to form.

Will you comment more on what role diffraction played in discovering this star and what it is made of? Dispersion, refraction, diffraction, reflection,.. where is Mrs. Roth? I'd Love to hear her comments as well. We had a blast discussing this with her in our labs,.. and you at the board!

4. Well, first it would depend on diffraction caused by the telescope, rather than diffraction caused by the eye. My understanding is that the limitations of atmospheric distortions are a limiting factor in astronomical observation (unless this star was discovered using the Hubble Space Telescope). As far as what it is made of, that would depend on spectroscopy more than imaging. See chapter 14 of Intermediate Physics for Medicine and Biology about spectroscopy. That is about all I know regarding Leo.